On the structure of rotation sets in hyperbolic surfaces

Searching for a relation between the genus g >= 2 of a closed oriented surface and the possible geometries for homological rotation sets of its maps, we prove that this invariant for Smale diffeomorphisms is given by a union of at most 2(5g-3) convex sets, all of them containing zero. The classical theory of hyperbolic dynamics allows then to extend this bound to a C-0-open and dense set of homeomorphisms, suggesting this to be a general fact. Examples showing the sharpness for this asymptotic order are provided.

Automated summary

In this study, we establish the relationship between the genus and possible geometries for homological rotation sets of maps on closed oriented surfaces with a genus g >= 2. We demonstrate that this invariant for Smale diffeomorphisms can be described as the union of at most 2(5g-3) convex sets, all containing zero. By utilizing the theory of hyperbolic dynamics, we extend this bound to a C-0-open and dense set of homeomorphisms, indicating its general validity. Additionally, we provide examples that illustrate the sharpness of this asymptotic order.